isosctrlem1 - Metamath Proof Explorer

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Theorem isosctrlem1 27128

Description: Lemma for isosctr 27131. (Contributed by Saveliy Skresanov, 30-Dec-2016.)

**Assertion**
Ref	Expression
isosctrlem1	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

**Proof of Theorem isosctrlem1**
Step	Hyp	Ref	Expression
1		ax-1cn 11239	. . . . . . 7 ⊢ 1 ∈ ℂ
2		subcl 11537	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
3	1, 2	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ)
4	3	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
5		subeq0 11565	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
6	5	notbid 321	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴))
7	1, 6	mpan 703	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴))
8	7	biimpar 483	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
9	8	neqned 2963	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
10	4, 9	logcld 26880	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
11	10	imcld 15342	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
12	11	3adant2 1149	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
13	3	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
14	9	3adant2 1149	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
15		releabs 15469	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴))
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴))
17		breq2 5107	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1))
18	17	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1))
19	16, 18	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1)
20		recl 15257	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
21	20	recnd 11318	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ)
22	21	subidd 11638	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
23	22	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
24		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 𝐴 ∈ ℂ)
25	24	recld 15341	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ∈ ℝ)
26		1red 11290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 1 ∈ ℝ)
27		simpr 490	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ≤ 1)
28	25, 26, 25, 27	lesub1dd 11913	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
29	23, 28	eqbrtrrd 5129	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 0 ≤ (1 − (ℜ‘𝐴)))
30	19, 29	syldan 603	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴)))
31		resub 15274	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)))
32		re1 15301	. . . . . . . . . . 11 ⊢ (ℜ‘1) = 1
33	32	oveq1i 7422	. . . . . . . . . 10 ⊢ ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴))
34	31, 33	eqtrdi 2812	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
35	1, 34	mpan 703	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
36	35	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
37	30, 36	breqtrrd 5133	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴)))
38	37	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 0 ≤ (ℜ‘(1 − 𝐴)))
39		neghalfpirx 26777	. . . . . 6 ⊢ -(π / 2) ∈ ℝ^*
40		halfpire 26775	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
41	40	rexri 11348	. . . . . 6 ⊢ (π / 2) ∈ ℝ^*
42		argrege0 26921	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
43		iccleub 13513	. . . . . 6 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^* ∧ (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
44	39, 41, 42, 43	mp3an12i 1494	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
45	13, 14, 38, 44	syl3anc 1398	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
46		pirp 26772	. . . . 5 ⊢ π ∈ ℝ⁺
47		rphalflt 13132	. . . . 5 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
48	46, 47	ax-mp 5	. . . 4 ⊢ (π / 2) < π
49	45, 48	jctir 530	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π))
50		pire 26765	. . . . . . 7 ⊢ π ∈ ℝ
51	50	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → π ∈ ℝ)
52	51	rehalfcld 12574	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (π / 2) ∈ ℝ)
53		lelttr 11381	. . . . 5 ⊢ (((ℑ‘(log‘(1 − 𝐴))) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
54	11, 52, 51, 53	syl3anc 1398	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
55	54	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
56	49, 55	mpd 16	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π)
57	12, 56	ltned 11427	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11522 -cneg 11523 / cdiv 11954 2c2 12378 ℝ⁺crp 13101 [,]cicc 13460 ℜcre 15244 ℑcim 15245 abscabs 15381 πcpi 16212 logclog 26864

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-log 26866

This theorem is used by: isosctrlem2 27129

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